A Johnson-type bound is a positive number $J$ depending on the distance,
the blocklength and the cardinalities of the Alphabets constituting the
code. It garanties that a ``small" number of codewords are in any sphere
of radius $J$. By ``small" number, we mean a number of codewords which is
linear in the code blocklength and the cardinality of the code.
But it turns out that in our case, the Johnson-type bound for number
fields codes depends only on the code blocklength and its minimal
distance, and that, ``small" means polynomial in
$\sum_{i = 1}^n \log \Nm(\p_i)$.

In this section, we show that the Johnson-type bound of
\cite[Section 7.6.1]{guruswami_phd} remains valid for number field codes.
For any prime ideal $\p\subset\OK$, the quotient $\OK/\p$ is a finite
field. Thus the $i$'th symbol of a codeword comes from an alphabet of size
$\Nm(\p_i) = |\OK/\p_i|$ and Theorem~7.10 of \cite{guruswami_phd} can be
applied. For the sake of completness, we state the theorem.
We denote by $[q_i]$ where $q_i$ is an integer the set
$\{ 1,2,3,\ldots,q_i \}$.

\begin{theorem}
Let $C$ be a code of blocklength $n$ with the $i$'th symbol comming from an
alphabet of size $q_i$, for $1 \leq i \leq n$. Let the distance
$D_{\alpha}$ of the code be measured according to a weighting vector
$\alpha = (\alpha_1,\ldots,\alpha_n)$. For a weighting vector
$\beta = (\beta_1,\ldots,\beta_n)$ and a received word
$r = (r_1,\ldots,r_n) \in [q_1] \times [q_2] \times \ldots \times [q_n]$,
define the set $S_{\beta}(r,W)$ to consist of all words with weighted
$\beta$-weighted agreement with $r$ at least $W$. Then, for all $r$, the
set $S_{\beta}(r,W)$ has at most $\left(2\sum_{i=1}^n q_i\right)$
codewords from $C$, provided that:
$$
W \geq \left[ \left( \sum_{i=1}^n \alpha_i - D_{\alpha} \right)
       \sum_{i=1}^n \frac{\beta_i^2}{\alpha_i} \right]^{1/2},
$$
and at most $L$ codewords from $C$, provided that
$$
W \geq \left[ \left( \sum_{i=1}^n \alpha_i - D_{\alpha} 
                     + \frac{D_{\alpha}}{L} \right)
       \sum_{i=1}^n \frac{\beta_i^2}{\alpha_i} \right]^{1/2}.
$$
\end{theorem}

\begin{proof}
\cite[Proof of theorem 7.10, page 163]{guruswami_phd}.
\end{proof}

Let $t$ be the least positive integer such that
$$
\prod_{i=1}^t \Nm(\p_i) > \left(\frac{2B}{d}\right)^d,
$$
where $d = [K:\Q]$ and let
$$
T = \prod_{i=1}^t \Nm(\p_i).
$$
Then, by \cite[Lemma~12]{guruswami_nb_fld}, the minimal hamming distance
of the number fields code is at least $n - t + 1$. As a consequence the
$(1) = (1,\ldots,1)$-weighted distance is at least $n - t + 1$ or
$D_{(1)} \geq n - t + 1$ and:

\begin{lemma}
The $w_{\log} = (\log\Nm(\p_1),\ldots,\log\Nm(\p_n))$-weighted
distance of the number field code is at least $\log N - \log T$.
\end{lemma}

\begin{proof}
Let $m_1$ and $m_2$ be two codewords. Let
$\Ec = \{ i \leq n | m_1 - m_2 \not= 0 \mod \p_i\}$,
$\Ac = \{ i \leq n | i \not\in \Ec \}$,
$P_{\Ec} = \prod_{i \in \Ec} \Nm(\p_i)$ and
$P_{\Ac} = \prod_{i \in \Ac} \Nm(\p_i) = N/P_{\Ec}$.
Then $e = m_1 - m_2 \in \prod_{i \in \Ac} \p_i$, therefore
$\Nm(e) \geq P_{\Ac}$, but $\left( \frac{2B}{d} \right)^d \geq \Nm(e)$
and $T \geq P_{\Ac} = N/P_{\Ec}$, hence the result.
\end{proof}

Using the above bounds for the distance, we can state the following
conditions under which number fields codes list decoding algorithms
will output a ``small" (as defined above) list of codewords:
\begin{equation}
\label{eq:johnson_hamming}
\sum_{i = 1}^n a_i > \sqrt{(t + \varepsilon)n}
\end{equation}
\begin{equation}
\label{eq:johnson_log}
\sum_{i = 1}^n a_i \log\Nm(\p_i) >
\sqrt{(\log T + \varepsilon ) \log N}
\end{equation}
Where $\varepsilon$ is a nonnegative real number.
We may expect our list decoding algorithm to decode at least
a number of errors given by the bound of equations
\ref{eq:johnson_hamming} and
\ref{eq:johnson_log}.

